Example 4.6.8.12. If $K = \Delta ^0$, then the suspension $\Sigma (K)$ can be identified with $\Delta ^1$. In this case, the simplicial path category $\operatorname{Path}[ \Sigma (K) ]_{\bullet }$ can be identified with the ordinary category $[1]$, and the simplicial set $\Phi (K)$ is isomorphic to $\Delta ^{0}$.
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